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Question 4 Consider a spring pendulum depicted in Figure 1 in which a mass M is free to move along a horizontal line and is connected to a fixed point by an ideal, massless spring with spring constant k. A point mass m is fixed to the other end of a massless, in- extensible rod of length 1. The motions of masses M and m are confined to the same vertical plane. y=0 k ४४४४४ (a + x.0) M (₁-3₁) Figure 1: The spring pendulum.

Question 4 Consider a spring pendulum depicted in Figure 1 in which a mass M is free to move along a horizontal line and is connected to a fixed point by an ideal, massless spring with spring constant k. A point mass m is fixed to the other end of a massless, in- extensible rod of length 1. The motions of masses M and m are confined to the same vertical plane. y=0 k ४४४४४ (a + x.0) M (₁-3₁) Figure 1: The spring pendulum.

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Question 4 Consider a spring pendulum depicted in Figure 1 in which a mass M is free to move along a horizontal line and is connected to a fixed point by an ideal, massless spring with spring constant k. A point mass m is fixed to the other end of a massless, in- extensible rod of length I. The motions of masses M and m are confined to the same vertical plane. (a + x.0) I k y=0 } = ૦ | - ૪૪૪ ૪ ૪ M l (19₁) Figure 1: The spring pendulum. a. Write down the Lagrangian of the system in terms of the extension of the spring, x, and the angular displacement of the pendulum. b. Apply Lagrange's equations to find the equations of motion of the system. c. Now consider the approximation of small oscillations about the equilibrium po- sition. Show that the equations of motion reduce to: [M+m]ï + mlő + kx=0, and mlï+ml²Ö + mgle = 0. Justify carefully each approximation you make. d. For the case M = m show that the angular frequencies, w, of small amplitude normal mode oscillations about the equilibrium position satisfy the equation k w² - w² (²/7/4 + 1) + 39th 2 = gk 0 Im
Transcribed Image Text:Question 4 Consider a spring pendulum depicted in Figure 1 in which a mass M is free to move along a horizontal line and is connected to a fixed point by an ideal, massless spring with spring constant k. A point mass m is fixed to the other end of a massless, in- extensible rod of length I. The motions of masses M and m are confined to the same vertical plane. (a + x.0) I k y=0 } = ૦ | - ૪૪૪ ૪ ૪ M l (19₁) Figure 1: The spring pendulum. a. Write down the Lagrangian of the system in terms of the extension of the spring, x, and the angular displacement of the pendulum. b. Apply Lagrange's equations to find the equations of motion of the system. c. Now consider the approximation of small oscillations about the equilibrium po- sition. Show that the equations of motion reduce to: [M+m]ï + mlő + kx=0, and mlï+ml²Ö + mgle = 0. Justify carefully each approximation you make. d. For the case M = m show that the angular frequencies, w, of small amplitude normal mode oscillations about the equilibrium position satisfy the equation k w² - w² (²/7/4 + 1) + 39th 2 = gk 0 Im
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